\[ \begin{align} D_M &= ((\vec{x} - \vec{y})' {\rm \bf C}^{-1} (\vec{x} - \vec{y}) )^{\frac{1}{2}},\\ D_F (f, g) &= \inf_{\alpha, \beta} \max_{t\in[0,1]} d(f(\alpha(t)), g(\beta(t))),\\ D_{LSML}(x_i, x_j) &= \sqrt{(x_i - x_j)^TWW^T(x_i - x_j)}.\\ \end{align} \]

SMOCK data (Herngreen et al. 1994)

We will need the following libraries:

library(mice)
library(tidyverse)
library(brokenstick)
library(matlib)
library(ggplot2)
library(ggrepel)
library(Rcpp)
library(latex2exp)
library(plotly)

p <- ggplot(donors, aes(x = predictive_distance,
    y = MD, color = factor(pmm), shape = factor(md),
    alpha = factor(all))) + scale_alpha_discrete(range = c(0.4,
    1)) + geom_point(color = ifelse(donors$plotname %in%
    matches_pmm, "red", "black")) + geom_point(data = donors %>%
    filter(plotname %in% matches_blended),
    pch = 21, size = 4, colour = "green") +
    geom_label_repel(aes(label = three, segment.size = 0.1),
        data = donors %>%
            filter(three %in% matches_three),
        min.segment.length = 0, label.padding = 0.1,
        size = 2, max.overlaps = 20, color = "black") +
    labs(x = TeX("Predictive Distance"),
        y = TeX("Mahalanobis Distance"),
        title = "Distance of each donor") +
    theme(legend.position = "none")